Small samples lie
Hygiene surveys are often 6–12 measurements. A simple average or a “highest result” treats a handful of readings as the whole truth and hides how uncertain that really is.
Bayesian Exposure Statistics
Exposure decisions you can defend — proven by study.
HazNode turns a handful of air-monitoring results into rigorous, presentation-ready statistics. It runs the peer-reviewed WebExpo Bayesian models — the same method behind the Expostats toolbox, developed by a team led from the Université de Montréal — then put our own production engine through a rigorous validation study to prove it.
Peer-reviewed method (IRSST R-1065)
Validated across 9,640 analyses
Censored-data ready
Built on research from
The WebExpo / Expostats Bayesian methodology was developed by a research team led from the Université de Montréal and published by the IRSST.
Peer-reviewed
Published methodology — IRSST report R-1065 (Lavoué et al., 2020), peer-reviewed under IRSST policy.
Open methodology
The same open WebExpo models behind the public Expostats toolbox — no black box, no proprietary maths.
Validated in production
Our engine reproduces the published reference results and stays honestly calibrated at every sample size.
Why Bayesian
A spreadsheet average can't defend a decision
Occupational exposure data are noisy, lognormal, and almost always scarce. The classic ways of summarising them quietly throw away the one thing a client and a regulator actually need: how confident you are.
Non-detects get fudged
Results below the laboratory limit are routinely replaced with a guess (LOD/2). Bayesian censored likelihoods model them properly instead of inventing a number.
A pass/fail hides the risk
A yes/no verdict tells a client nothing about how confident you are. A probability — “86.8% chance the limit is exceeded” — is defensible and honest.
Three risk lenses · one verdict
Every survey, judged three ways
HazNode assesses each exposure group against three independent Bayesian criteria, each expressed as a probability and compared to the acceptability threshold ψ = 30%. A group is only “acceptable” when its probability of breaching the criterion sits below that line.
Exceedance fraction
P(EF ≥ 5%) < 30%
The probability that more than 5% of all exposures exceed the limit. Answers “how often is the OEL breached?” — the lens regulators and most modern guidance lean on.
95th percentile
P(P95 ≥ OEL) < 30%
How high the worst-case days reach. The model estimates the 95th percentile of the exposure profile and tests the probability it sits at or above the occupational exposure limit.
Arithmetic mean
P(AM ≥ OEL) < 30%
What the long-run average exposure looks like — the lens that matters for cumulative-dose contaminants. Tested as a probability, not a single point estimate.
Proven by study
We didn't just implement the science. We tested it.
The WebExpo authors left two practical questions open: how long should the Monte-Carlo run be, and is the stated uncertainty honest across sample sizes? We answered both. Our own validation study put HazNode's production engine through a known-truth gauntlet — and it reproduced WebExpo's published reference results exactly.
9,640
independent analyses
run against the live production engine
2 billion+
MCMC iterations
completed with a 0% error rate
89–100%
calibration coverage
against a nominal 90% target
~3.4 s
median compute
per analysis at production settings
What the study found
Faithfully reproduces WebExpo’s published reference dataset — same geometric mean, spread and exceedance, inside the published ranges.
Credible intervals are well calibrated at every sample size (89–100% coverage vs a nominal 90%), and slightly conservative — never overconfident — on the smallest samples.
Run-to-run numerical noise falls as ~1/√(iterations) and plateaus beyond ~500,000 — so the production policy is set, not guessed.
An evidence-based policy of 250,000 iterations for n < 3 and 75,000 otherwise gives consistent precision at a sub-second to ~1-second cost per analysis.
Precision is not certainty
More iterations sharpen the numerical precision of a reported figure — not the statistical certainty of the estimate. A small sample has an irreducibly wide credible interval; we render it precisely and report it honestly, rather than make it look more confident than the data allow. That distinction holds in every number HazNode shows you.
Contact us and we'll send you the full validation paper.
Method: WebExpo SEG lognormal model (JAGS, informedvar prior), Lavoué et al., IRSST R-1065 (2020). Validation: HazNode engineering working paper, June 2026 — 9,640 analyses across reference, precision and known-truth calibration studies, 0% error rate.
One-of-a-kind reporting
You decide exactly what the client sees
No other exposure-statistics tool gives you this. Toggle any section on or off — every chart, every risk lens, your commentary, the full methodology appendix — preview it live, and export a clean, presentation-ready PDF on your own letterhead.
Create PDF Report
Core statistics
Measurements
KPI cards
Statistics table
Q-Q plot
Box & whiskers
Risk lens findings
Exceedance fraction
95th percentile
Arithmetic mean
Detailed drilldown
Risk meter
Estimates
Sequential chart
Density chart
Risk band chart
Narrative
Hygienist commentary
Methodology & assumptions
Create PDF
Straight off the page · one of many report sections
Bayesian Statistical Report
Generated by HazNode
B.1
Exceedance fraction
criterion: P(EF ≥ 5%) < ψ
Problematic — P = 86.8%
The probability that more than 5% of measurements exceed the OEL is 86.8%, above the ψ = 30% acceptability threshold. The point estimate places 14.2% of measurements above the OEL with a 90% credible interval of [2.8%–40.0%]. This indicates likely overexposure on a portion of working days.
Sequential plot
Density plot
Risk band plot
Posterior probability that EF ≥ 5%
p = 86.8%
ψ = 30%
0%
25%
50%
75%
100%
◀ Acceptable
Problematic ▶
Point estimate
EF = 14.2%
90% CI [2.8%–40.0%]
UCL₇₀
20.9%
conservative estimate
P(EF ≥ 5%)
86.8%
target < 30%
Section-level control
Turn any block on or off — measurements, KPI cards, Q-Q and box plots, each of the three risk lenses, and every underlying chart.
Live preview
See the exact report update as you toggle, so what you approve on screen is what prints.
Your commentary, your brand
Add hygienist commentary and ship it on your letterhead — “Generated by HazNode”, prepared by you.
Defensible by default
Keep the methodology appendix in for audit and expert-witness work, or strip it out for a one-page client summary.
Questions
Bayesian exposure statistics, explained
What is Bayesian exposure statistics?
Bayesian exposure statistics interpret occupational monitoring data by fitting a lognormal exposure profile and expressing the result as probabilities — for example, the probability that more than 5% of exposures exceed the occupational exposure limit (OEL). Unlike a simple average, it states how certain the conclusion is, handles non-detect (censored) results correctly, and remains defensible on the small samples that are typical in industrial hygiene.
What model does HazNode use?
HazNode implements the WebExpo Bayesian lognormal models published by the IRSST (report R-1065, Lavoué et al., 2020) — the same peer-reviewed methodology behind the Expostats toolbox, developed by a team led from the Université de Montréal. The statistical model and priors are exactly those published by WebExpo; HazNode runs them as a production engine.
Is HazNode’s engine validated?
Yes. We ran our production engine through an in-house validation study of 9,640 analyses and more than two billion MCMC iterations at a 0% error rate. It reproduces WebExpo’s published reference results, and its credible intervals are well calibrated across every sample size (89–100% coverage against a nominal 90% target), sitting slightly conservative on the smallest samples.
How many MCMC iterations does it run?
HazNode uses an evidence-based production policy of 250,000 iterations for very small samples (n < 3) and 75,000 otherwise. The validation study showed run-to-run numerical noise falls roughly as 1/√(iterations) and plateaus beyond about 500,000, so these settings deliver consistent, high-precision reported figures at a sub-second to roughly one-second compute cost per analysis.
Does more iterations mean more certainty?
No — and HazNode is deliberate about this. More iterations improve the numerical precision of the reported figure, not the statistical certainty of the estimate. A small sample has an irreducibly wide credible interval; running it longer renders that interval precisely, it does not narrow it. Honest, sometimes wide, uncertainty is the point.
Can I choose what goes in the report?
Yes. HazNode’s report builder lets you toggle every section on or off — measurements, KPI cards, the three risk lenses, individual charts, hygienist commentary and the methodology appendix — then export a clean, presentation-ready PDF on your own letterhead. You decide exactly what the client sees.
See defensible exposure statistics on your own data
Book a live walkthrough and we'll run the Bayesian engine and the report builder on a real sample group with you.
